81 research outputs found
Formal groups arising from formal punctured ribbons
We investigate Picard functor of a formal punctured ribbon. We prove that
under some conditions this functor is representable by a formal group scheme.
Formal punctured ribbons were introduced in arXiv:0708.0985.Comment: 42 pages, minor change
Coherent analogues of matrix factorizations and relative singularity categories
We define the triangulated category of relative singularities of a closed
subscheme in a scheme. When the closed subscheme is a Cartier divisor, we
consider matrix factorizations of the related section of a line bundle, and
their analogues with locally free sheaves replaced by coherent ones. The
appropriate exotic derived category of coherent matrix factorizations is then
identified with the triangulated category of relative singularities, while the
similar exotic derived category of locally free matrix factorizations is its
full subcategory. The latter category is identified with the kernel of the
direct image functor corresponding to the closed embedding of the zero locus
and acting between the conventional (absolute) triangulated categories of
singularities. Similar results are obtained for matrix factorizations of
infinite rank; and two different "large" versions of the triangulated category
of relative singularities, corresponding to the approaches of Orlov and Krause,
are identified in the case of a Cartier divisor. A version of the
Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix
factorizations and disproven for locally free matrix factorizations of finite
rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the
Serre-Grothendieck duality theorems for matrix factorizations are established,
and pull-backs and push-forwards of matrix factorizations are discussed at
length. A number of general results about derived categories of the second kind
for CDG-modules over quasi-coherent CDG-algebras are proven on the way.
Hochschild (co)homology of matrix factorization categories are discussed in an
appendix.Comment: LaTeX 2e with pb-diagram and xy-pic; 114 pages, 13 commutative
diagrams. v.8: new sections 2.10, 3.1 and 3.7 inserted; v.9: appendix B
added, remarks inserted in sections 2.10 and 2.7, section 1.8 expanded; v.10:
new section 3.3 inserted, the whole paper has two authors now; v.11: small
corrections, additions, and improvements -- this is intended as the final
versio
Infinite-dimensional supermanifolds over arbitrary base fields
In his recent investigation of a super Teichm\"uller space, Sachse (2007),
based on work of Molotkov (1984), has proposed a theory of Banach
supermanifolds using the `functor of points' approach of Bernstein and Schwarz.
We prove that the the category of Berezin-Kostant-Leites supermanifolds is
equivalent to the category of finite-dimensional Molotkov-Sachse
supermanifolds. Simultaneously, using the differential calculus of
Bertram-Gl\"ockner-Neeb (2004), we extend Molotkov-Sachse's approach to
supermanifolds modeled on Hausdorff topological super-vector spaces over an
arbitrary non-discrete Hausdorff topological base field of characteristic zero.
We also extend to locally k-omega base fields the `DeWitt' supermanifolds
considered by Tuynman in his monograph (2004), and prove that this leads to a
category which is isomorphic to the full subcategory of Molokov-Sachse
supermanifolds modeled on locally k-omega spaces.Comment: 36 pages; minor corrections, expanded introductio
Intersections of translated algebraic subtori
We exploit the classical correspondence between finitely generated abelian
groups and abelian complex algebraic reductive groups to study the intersection
theory of translated subgroups in an abelian complex algebraic reductive group,
with special emphasis on intersections of (torsion) translated subtori in an
algebraic torus.Comment: 20 pages; accepted for publication by the Journal of Pure and Applied
Algebr
Reflexive representability and stable metrics
It is well-known that a topological group can be represented as a group of
isometries of a reflexive Banach space if and only if its topology is induced
by weakly almost periodic functions (see
\cite{Shtern:CompactSemitopologicalSemigroups},
\cite{Megrelishvili:OperatorTopologies} and
\cite{Megrelishvili:TopologicalTransformations}). We show that for a metrisable
group this is equivalent to the property that its metric is uniformly
equivalent to a stable metric in the sense of Krivine and Maurey (see
\cite{Krivine-Maurey:EspacesDeBanachStables}). This result is used to give a
partial negative answer to a problem of Megrelishvili
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